Posets, clique graphs and their homotopy type

To any finite poset P we associate two graphs which we denote by Ω ( P ) and ℧ ( P ) . Several standard constructions can be seen as Ω ( P ) or ℧ ( P ) for suitable posets  P , including the comparability graph of a poset, the clique graph of a graph and the 1-skeleton of a simplicial complex. We interpret graphs and posets as simplicial complexes using complete subgraphs and chains as simplices. Then we study and compare the homotopy types of Ω ( P ) , ℧ ( P ) and P . As our main application we obtain a theorem, stronger than those previously known, giving sufficient conditions for a graph to be homotopy equivalent to its clique graph. We also introduce a new graph operator H that preserves clique-Hellyness and dismantlability and is such that H ( G ) is homotopy equivalent to both its clique graph and the graph G .